The objective of this research was to develop a methodology for optimizing multilayer-perceptron-type neural networks by evaluating the effects of three neural architecture parameters, namely, number of hidden layers (HL), neurons per hidden layer (NHL), and activation function type (AF), on the sum of squares error (SSE). The data for the study were obtained from quality parameters (physicochemical and microbiological) of milk samples. Architectures or combinations were organized in groups (G1, G2, and G3) generated upon interspersing one, two, and three layers. Within each group, the networks had three neurons in the input layer, six neurons in the output layer, three to twenty-seven NHL, and three AF (tan-sig, log-sig, and linear) types. The number of architectures was determined using three factorial-type experimental designs, which reached 63, 2 187, and 50 049 combinations for G1, G2 and G3, respectively. Using MATLAB 2015a, a logical sequence was designed and implemented for constructing, training, and evaluating multilayer-perceptron-type neural networks using parallel computing techniques. The results show that HL and NHL have a statistically relevant effect on SSE, and from two hidden layers, AF also has a significant effect; thus, both AF and NHL can be evaluated to determine the optimal combination per group. Moreover, in the three study groups, it is observed that there is an inverse relationship between the number of processors and the total optimization time.

December
Multilayer perceptron architecture optimization using parallel computing techniques
Wilson Castro 0 1
Jimy Oblitas 1
Roberto Santa-Cruz 1
Himer Avila-George 1
0 Facultad de IngenierÂõa, Universidad Privada del Norte, Cajamarca, Peru, 2 Centro de Investigaciones e Innovaciones de la Agroindustria Peruana, Amazonas, Peru, 3 Facultad de IngenierÂõa de Sistemas y Mec aÂnica EleÂ ctrica, Universidad Nacional Toribio RodrÂõguez de Mendoza, Chachapoyas, Peru, 4 Escuela de Doctorado, Departamento de TecnologÂõa de Alimentos, Universidad de Lleida , Lleida , Spain , 5 Unidad de Transferencia Tecnol oÂgica Tepic, CONACYT-CICESE , Tepic, Nayarit , Mexico
1 Editor: Alexandre G. de Brevern, UMR-S1134, INSERM, UniversiteÂ Paris Diderot , INTS , FRANCE
The objective of this research was to develop a methodology for optimizing multilayer-perceptron-type neural networks by evaluating the effects of three neural architecture parameters, namely, number of hidden layers (HL), neurons per hidden layer (NHL), and activation function type (AF), on the sum of squares error (SSE). The data for the study were obtained from quality parameters (physicochemical and microbiological) of milk samples. Architectures or combinations were organized in groups (G1, G2, and G3) generated upon interspersing one, two, and three layers. Within each group, the networks had three neurons in the input layer, six neurons in the output layer, three to twenty-seven NHL, and three AF (tan-sig, log-sig, and linear) types. The number of architectures was determined using three factorial-type experimental designs, which reached 63, 2 187, and 50 049 combinations for G1, G2 and G3, respectively. Using MATLAB 2015a, a logical sequence was designed and implemented for constructing, training, and evaluating multilayer-perceptron-type neural networks using parallel computing techniques. The results show that HL and NHL have a statistically relevant effect on SSE, and from two hidden layers, AF also has a significant effect; thus, both AF and NHL can be evaluated to determine the optimal combination per group. Moreover, in the three study groups, it is observed that there is an inverse relationship between the number of processors and the total optimization time.
-
Data Availability Statement: All relevant data are
within the paper and its Supporting Information
files.
Funding: The author(s) received no specific
funding for this work.
Competing interests: The authors have declared
that no competing interests exist.
Introduction
In applied research, it is common to encounter situations in which it is necessary to estimate
the behavior of a variable as a function of one or many predictor variables. Traditionally, the
solution is provided by statistical regression models for prediction problems, discriminant
analysis, or logistic regression models [1, 2]. A group of techniques known as artificial
intelligence offers other options, including artificial neural networks, genetic algorithms, and fuzzy
logic, among others, which are suitable for solving complex problems [3±6].
Artificial neural networks (ANNs), which are non-linear models inspired by the neural
architecture of the brain, were developed in an attempt to model the learning capacity of
biological neural systems [7]. A typical ANN architecture known as multilayer perceptron (MLP)
contains a series of layers, composed of neurons and their connections. An artificial neuron
has the ability to calculate the weighted sum of its inputs and then applies an activation
function to obtain a signal that will be transmitted to the next neuron.
The development of MLP networks has two main problems: architecture optimization and
training. The definition of architecture is a very relevant point because a lack of connections
can make the network incapable of solving the problem of insufficient adjustable parameters,
whereas an excess of connections may cause an over-fitting of the training data [8].
Consequently, training MLP networks for large datasets is very time consuming [9].
Determination of the optimal architecture is a constant goal in research papers [10±12],
which attempt to minimize an objective function, mean squared error or prediction residual
sum of squares errors and avoid the oversize of the network; the method used in these research
works is trial and error. However, the trial and error method limits the capacity of analyzable
architectures and reduces the likelihood of finding an optimal architecture, particularly if we
have a large number of possible architectures. Different approaches have been proposed to
optimize the architecture of an MLP network, for example, back-propagation [13], genetic
algorithms [14], ant colony [15], bee swarm [16], and Tabu search [17], among others.
Similarly, different approaches have been proposed to manage the expensive training phase, for example, the use of multicore CPU [18±20], cloud computing [21] and hybrid algorithms [8, 15], among others.
In this paper, we focus on the problem of constructing an optimal multilayer perceptron
network architecture. Given the popularity and easy access to equipment with large multi-core
multiprocessing or GPU capabilities, which enable the parallel calculation of multiple
operations, a comprehensive approach to finding the optimal architecture of a multilayer perceptron
network is proposed. Four versions of the proposed approach, i.e., sequential, multi-core,
GPU and a hybrid algorithm, are introduced.
The objectives of this research are as follows: (a) propose a methodology for optimizing multilayer-perceptron-type neural networks, (b) evaluate the effects of the different structural parameters on the sum of squares error, and (c) evaluate the performance of the optimization process using parallel computing techniques.
Materials and methods
Materials
As a biological material, 252 milk samples from Holstein cows (40 cc/sample) were collected.
Sampling was conducted between July and August 2014. The milk samples were not collected
exclusively for this work; they were provided by employees of NestleÂ S. A. from 12 centers
located in the countryside of Cajamarca, Peru. Fig 1 shows the map of the Cajamarca region,
where the twelve points of milk collection are indicated, and Table 1 shows the geographic
coordinates of each of the twelve points.
NestleÂ is an internationally recognized company that is very committed to the welfare of
animals, as indicated by their strict guidelines for animal care [22].
Computing system
A computer called Perseo was used for the experiment, which is a shared-memory multipro
cessor. Perseo is part of the network of computer equipment belonging to the Centro Nayarita
de Innovación y Transferencia de Tecnología A.C, Mexico.
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Fig 1. Cajamarca region map.
3 / 17
The main characteristics of Perseo are as follows:
· CPU cores = 24
· RAM Memory = 32 GB
· Graphic card = NVIDIA Quadro K4200
· Operating System = Ubuntu 14.04
The software used for implementing the logical sequences was MATLAB version 2015a.
Experimental methodology
Obtaining training and validation data. Samples were packed in sampling bottles and transported to the lab of the Nestle plant, located in the city of Cajamarca, where they were characterized as shown in Table 2.
The data obtained from each sample were divided into three input values: density (Dn), oxidation-reduction potential (Rd), and potential of hydrogen (pH). Moreover, six output parameters were defined: proteins (Pr), lactose (Lc), total solids (Ts), solids-fat (Sf), solids-non-fat (Snf), and minerals (Mn).
Condensed architecture for multilayer perceptrons. Fig 2 shows the proposed
multilayer perceptron architecture, which is based on the following works [27±29].
Method
Lactodensimeter (AOAC 925.22)
Reaction time to methylene blue
Potentiometer
Infrared spectroscopy (NTP 202.130:1998)
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Range
3
6
[1-3]
[
3-27
]
[1-3]
The structural parameters to evaluate and their ranges were established in accordance with [30]; see Table 3.
After modifying the number of hidden layers, inter-spacing from one to three hidden lay
ers, different architectural groups were generated, as shown in Fig 3.
For the bias parameter values and weightings, prior initiation to zero was determined during the optimization process.
Experimental designs and generation of combinations per group. Table 4 shows the
experimental designs used in this research work; the architectures were generated per group
and evaluated using factorial designs without repetition.
The position of each element within the groups and the number of combinations (treatments) per experimental design are detailed in Table 5.
Combinations per group according to the proposed designs were generated using the statistical software Statgraphics Centurion XVI.
Perceptrons per group of combinations: Generation and evaluation. To create, train
and simulate MLP-type networks, MATLAB's Neural Network Toolbox was used, particularly
the function newff, whose syntax is shown in Eq (1).
net  newff p; t; G i; j
1
where:
p: is the vector of input values.
t: is the vector of output values.
Gi,j is a combination per group (i: group; j: combination number).
The networks per group were evaluated by determining the sum of squares error (SSE)
using a logical sequence. This sequence was implemented in the mathematical software
MATLAB 2015a; see Fig 4.
Likewise, due to the large number of combinations per group and the high calculation time cost, the analysis sequence was implemented using profiling, vectorization and parallel computing techniques.
Initially, a sequential version of the proposed algorithm was developed (MLP-SEQ), which
was implemented using profiling and vectorization techniques. However, since the algorithm
has a comprehensive search approach, it is time consuming. Theoretically, the time required
by an algorithm to calculate the solution to a given problem using a single processor could be
linearly decreased by adding more processors. Following this idea, we developed the following
algorithms to attempt to reduce the processing time.
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Fig 3. Groups of neural architectures proposed for the study.
https://doi.org/10.1371/journal.pone.0189369.g003
6 / 17
Factors
Name
NHL1
AF1
AF2
NHL1
NHL2
AF1
AF2
AF3
NHL1
NHL2
NHL3
AF1
AF2
AF3
AF4
S  To
Tn
· NNTB-GPU development is based on MATLAB's Neural Network Toolbox, where the idea
is to take advantage of the computation based on the GPU and to process the search space in
parallel.
· NNTB-CPU is also based on MATLAB's Neural Network Toolbox, where the idea is to take
advantage of multi-CPU architectures to process data faster.
· NNTB-Hybrid merges the two previous approaches.
· PCTB-CPU is based on MATLAB's Parallel Computing Toolbox, proposing a distributed
computing approach (master-worker).
Analysis of evaluation times per group of combinations. Acceleration and efficiency are
some of the most important measurements for assessing the quality of the implementation of a
logical sequence (algorithm) on an architecture of multiprocessors [
31
]. The acceleration of a
logical sequence implemented in parallel executed using n processors is the ratio between the
time that it takes the best logical sequence implemented sequentially to be executed using a
single processor in a computer and the time that it takes the corresponding logical sequence
implemented in parallel to be executed on the same computer using n processors; see Eq (2).
where
Design
1HL2AF
2HL3AF
3HL4AF
Number of Treatments
2
Fig 4. Sequence for constructing, training, and evaluating networks.
S: Acceleration.
T0: Computing time with one processor.
Tn: Time with n processors.
If acceleration is normalized by dividing it by the number of processors, then efficiency is obtained; see Eq (3).
Z  Sn
n
3
where
η: Efficiency.
Sn: Speedup with one processor.
n: Number of processors.
Results and discussion
Training data
Parallel and sequential versions of the logical sequence shown in Fig 4 were developed. The obtained results were analyzed in terms of acceleration and efficiency.
The data collected during the milk analysis stage and later used in constructing, training, and evaluating the networks are shown in Table 6.
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Units
g/ml
hours
Ð
g/100 ml
g/100 ml
g/100 ml
g/100 ml
g/100 ml
g/100 ml
Combinations by groups of neural architectures
The first ten combinations by groups of neural architectures, which were used in the
construction of neural networks, are shown in Fig 5.
As can be appreciated, the architecture groups (G1, G2, and G3) differ in the number of
hidden layers (HL), which is a parameter that controls network accuracy. Within each group,
the differences are found in the number of neurons and the type of activation function. This
parameter has been researched as an element for neural network optimization in previous
works, such as those reported by [
33, 34
].
Architecture group analysis
The idea of applying parallel computing techniques in the area of neural networks has been used in the following research papers [18±21]. As shown, the use of distributed computing has
Fig 5. First ten combinations for G1, G2, and G3 and interpretation.
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been used mainly in the training phase of the neural network, which generally consumes
considerable computational resources when neural networks are vast and complex.
In this work, we developed three algorithms using the parallel techniques provided in
MATLAB's Neural Network Toolbox; these algorithms are NNTB-GPU, NNTB-CPU, and
NNTB-Hybrid. To compare their performance, we used the G2 dataset. The NNTB-GPU
algorithm used all the resources provided by a NVIDIA Quadro K4200 card; NNTB-CPU was run
using four processors; finally, NNTB-Hybrid was run using four processors and the graphics
card. The execution time, in seconds, of each algorithm was 3 028, 2 911, and 3 634,
respectively, with the algorithm NNTB-CPU obtaining better performance.
However, by running our sequential algorithm using the G2 dataset, the processing
lasted only 637 seconds. The reasons for this result could be the following: (1) the functions
that we use to parallelize the NNTB-GPU, NNTB-CPU and NNTB-Hybrid algorithms
obtain good performance when neural networks are complex, i.e., many input and
output neurons, many hidden layers, and so forth. However, as shown in Fig 3, the
neural networks that are processed in this research work do not typically have a complex
structure. For this reason, the three approaches previously mentioned spend more time in
establishing the parallel environment than in processing a particular neural network. In
fact, the problem that we are facing is to process many small neural networks to find the
optimal architecture.
Therefore, we developed the PCTB-CPU algorithm, which is based on the general functions
of the MATLAB PCTB. This algorithm uses a master-slave approach and creates a balanced
distribution of work among all available workers, i.e., the total number of architectures to be
tested is divided equally among the workers. The workers report their partial results to the
master, and the master is responsible for integrating all the information and submitting the
result. Following the previous example, the PCTB-CPU algorithm was tested using four
workers to process the dataset G2. This time, the duration was 218 seconds, which means that it
achieved 85% of the theoretical acceleration. Using this algorithm, the datasets G1, G2, and G3
were processed.
Regarding the resulting SSE by groups, according to factors, they are illustrated in Fig 6,
and they show that increasing the hidden layers reduces SSE dispersion, generating more
robust multilayer perceptrons; the results agree with the work of Garcia et al. [
35
] about the
relationship between the number of layers and the network efficiency, as well as with the
results obtained by Izquierdo et al. [
33
], who evaluated different structures until they
determined the optimal ones for their study conditions.
The analysis of the multifactorial variance for the SSE within each group according to NHL and AF is shown in Tables 7±9.
From the P-value, it is determined that the NHL has a statistically significant effect on the SSE in each group and that as of the second group (second hidden layer), the AF is added to it.
When evaluating the SSE and their relationship with the structural parameters per group,
images a, b, and c in Fig 7 are obtained. It can be observed that increasing the NHL reduced
the SSE until a minimal value is reached, and later increases cause an increase in the SSE. This
result is possibly due to the effect of over training, as explained by VelaÂsquez et al. [
36
] in their
study.
It is also observed that the SSE exhibits a different behavior for the various types of AFs and
the layer to which they connect. Therefore, the neural structure should be optimized by
minimizing the SSE according to the NHL and AF, determining the best combination of said
parameters. Table 10 presents the optimal values for the evaluated groups, as well as the
minimal SSEs.
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Fig 6. SSE for each group.
Optimization process times
During the experimental process, to measure the performance of the proposed logical sequence, three parameters (computing time, acceleration and efficiency) were used as references.
The times for determining the SSE in the various groups are shown in Fig 8. According to [33], the processing times may vary according to the characteristics of the equipment where
? Reliability level 99%.
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? Reliability level 99%.
the logical sequence is implemented. However, the trends shown by the results are similar to
those obtained in the work by [
37
] when the number of processors was successively increased
to the process.
From the analysis of the optimization process times, it is inferred that the implementation
requires a large calculation capacity for practical applications. In that sense, the speedup values
of the optimization process, Fig 9, show that there is an inverse relationship between the
number of processors and the total optimization times with a constant tolerance for the various
groups under study.
The information shown in Figs 8 and 9 are complemented with an analysis of the logical
sequence efficiency. Fig 10 shows the efficiency achieved by the proposed parallel algorithm
each time that it was tested using the G1, G2, and G3 treatment sets. For the G1 case, the
efficiency quickly decreases, which may be because more time is spent establishing the parallel
environment than processing treatment. However, for the G2 and G3 cases, the algorithm
reported an efficiency of over 70% even when using the maximum number of processors. This
result indicates that the proposed parallel algorithm could scale considerably for larger
experimental designs.
Fig 7. Interaction of structural parameters per group.
Conclusions
The optimal architecture of a multilayer-perceptron-type neural network may be achieved
using an analysis sequence of structural parameter combinations.
The number of hidden layers and the number of neurons per layer have statistically significant effects on the SSE. Likewise, the SSE shows a different behavior with respect to the various types of AF and the layer to which they connect.
The implementation of the logical sequence of the optimization is possible by applying parallel computing to the process, which reduces the process time and, depending on the number of processors, improves the performance.
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Fig 8. Calculation times per group and number of processors.
https://doi.org/10.1371/journal.pone.0189369.g008
Fig 9. Acceleration of optimization per groups.
https://doi.org/10.1371/journal.pone.0189369.g009
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Fig 10. Efficiency in the optimization of the groups.
https://doi.org/10.1371/journal.pone.0189369.g010
Supporting information
S1 File. Input data.
(ZIP)
16 / 17
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